Solve for $x$ : $ 6|x + 5| - 4 = -5|x + 5| + 2 $
Solution: Add $ {5|x + 5|} $ to both sides: $ \begin{eqnarray} 6|x + 5| - 4 &=& -5|x + 5| + 2 \\ \\ { + 5|x + 5|} && { + 5|x + 5|} \\ \\ 11|x + 5| - 4 &=& 2 \end{eqnarray} $ Add ${4}$ to both sides: $ \begin{eqnarray} 11|x + 5| - 4 &=& 2 \\ \\ { + 4} &=& { + 4} \\ \\ 11|x + 5| &=& 6 \end{eqnarray} $ Divide both sides by ${11}$ $ \dfrac{11|x + 5|} {{11}} = \dfrac{6} {{11}} $ Simplify: $ |x + 5| = \dfrac{6}{11}$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x + 5 = -\dfrac{6}{11} $ or $ x + 5 = \dfrac{6}{11} $ Solve for the solution where $x + 5$ is negative: $ x + 5 = -\dfrac{6}{11} $ Subtract ${5}$ from both sides: $ \begin{eqnarray} x + 5 &=& -\dfrac{6}{11} \\ \\ {- 5} && {- 5} \\ \\ x &=& -\dfrac{6}{11} - 5 \end{eqnarray} $ Change the ${ - 5}$ to an equivalent fraction with a denominator of $11$ $ x = - \dfrac{6}{11} {- \dfrac{55}{11}} $ $ x = -\dfrac{61}{11} $ Then calculate the solution where $x + 5$ is positive: $ x + 5 = \dfrac{6}{11} $ Subtract ${5}$ from both sides: $ \begin{eqnarray} x + 5 &=& \dfrac{6}{11} \\ \\ {- 5} && {- 5} \\ \\ x &=& \dfrac{6}{11} - 5 \end{eqnarray} $ Change the ${ - 5}$ to an equivalent fraction with a denominator of $11$ $ x = \dfrac{6}{11} {- \dfrac{55}{11}} $ $ x = -\dfrac{49}{11} $ Thus, the correct answer is $x = -\dfrac{61}{11} $ or $x = -\dfrac{49}{11} $.